Magnetic field

Current (I) flowing through a wire produces a magnetic field (B) around the wire. The field is oriented according to the right-hand rule.

For other senses of this term, see magnetic field (disambiguation).

In physics, a magnetic field is that part of the electromagnetic field that exists when there is a changing electric field. A changing electric field can be caused by the movement of an electrically charged object, as in an electric current; or a combination of the orbit of an electron around an atom and the spin of electrons themselves, as in a permanent magnet.

Definition

A magnetic field is the relativistic part of an electric field, as Einstein explained in 1905. When an electric charge is moving from the perspective of an observer, the electric field of this charge due to space contraction is no longer seen by the observer as spherically symmetric due to non-radial time dilation, and it must be computed using the Lorentz transformations. One of the products of these transformations is the part of the electric field which only acts on moving charges - and we call it the "magnetic field".

The quantum-mechanical motion of electrons in atoms produces the magnetic fields of permanent ferromagnets. Spinning charged particles also have magnetic moment. Some electrically neutral particles (like the neutron) with non-zero spin also have magnetic moment due to the charge distribution in their inner structure. Particles with zero spin never have magnetic moment.

A magnetic field is a vector field: it associates with every point in space a (pseudo-)vector that may vary through time. The direction of the field is the equilibrium direction of a magnetic dipole (like a compass needle) placed in the field.

The Lorentz transformation of a spherically-symmetric proper electric field E of moving electric charge (for example, electric field of an electron moving in a conducting wire) from charge's reference frame to non-moving observer's reference frame results in the following term which we can define or label as "magnetic field" and use the symbol B for it for the sake of mathematical simplicity (one symbol instead of seven):

$\mathbf {B} =\mathbf {v} \times {\frac {1}{c^{2}}}\mathbf {E}$

where

$\mathbf {v} \$ is velocity of the electric charge, measured in metres per second

$\times \$ indicates a vector cross product

c is the speed of light measured in metres per second

E is the electric field measured in newtons per coulomb or volts per metre

As seen from the definition, the unit of magnetic field is newton-second per coulomb-metre (or newton per ampere-metre) and is called the tesla. Like the electric field, the magnetic field exerts force on electric charge—but unlike an electric field, only on moving charge:

$\mathbf {F} =q\mathbf {v} \times \mathbf {B}$

where

F is the force produced, measured in newtons

$q\$ is electric charge that the magnetic field is acting on, measured in coulombs

$\mathbf {v} \$ is velocity of the electric charge $q\$, measured in metres per second

Because magnetic field is the relativistic product of Lorentz transformations, the force it produces is called the Lorentz force.

The force due to the magnetic field is different in different frames—moving magnetic field transforms partially or fully back into electric fields under Lorentz transformations. This results in Faraday's law of induction.

Magnetic field of current flow of charged particles

File:Charged-particle-drifts.gif

Charged particle drifts in a homegenous magnetic field. (A) No disturbing force (B) With an electric field, E (C) With an independent force, F (eg. gravity) (D) In an inhomgeneous magnetic field, grad H

Substituting into the definition of magnetic field

$\mathbf {B} =\mathbf {v} \times {\frac {1}{c^{2}}}\mathbf {E}$

the proper electric field of point-like charge (see Coulomb's law)

$\mathbf {E} ={1 \over 4\pi \epsilon _{0}}{q \over \mathbf {r} ^{2}}{\hat {\mathbf {r} }}={10^{-7}}{c^{2}}{q \over \ {r}^{2}}{\hat {\mathbf {r} }}$

results in the equation of magnetic field of moving charge, which is usually called the Biot-Savart law:

$\mathbf {B} =\mathbf {v} \times {\frac {\mu _{0}}{4\pi }}{\frac {q}{r^{2}}}\mathbf {\hat {r}}$

where

$q$ is electric charge, whose motion creates the magnetic field, measured in coulombs

v is velocity of the electric charge $q$ that is generating B, measured in metres per second

B is the magnetic field (measured in teslas)

Lorentz force on wire segment

Integrating the Lorentz force on an individual charged particle over a flow (current) of charged particles results in the Lorentz force on a stationary wire carrying electric current:

$F=IBl\,$

$n=20\,{\mbox{turns}}\,$

$I=5A\,$

$l=0.04m\,$

$B=0.8T\,$

where

F = force, measured in newtons

B = magnetic field, measured in tesla

l = length of wire, measured in metre

i = current in wire, measured in ampere

In the equation above, the current vector i is a vector with magnitude equal to the scalar current, i, and direction pointing along the wire in which the current is flowing.

Alternatively, instead of current, the wire segment l can be considered a vector.

The Lorentz force on a macroscopic current carrier is often referred to as the Laplace force.

Vector calculus

Separating the electric field of moving charge into stationary electric and magnetic components (as measured by a stationary observer)—which are usually labeled as E and B—replaces complex Einstein relativistic field transformation equations by more compact and elegant mathematical statements known as Maxwell's equations. The two of them that describe the magnetic component are:

$\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}$

$\nabla \cdot \mathbf {B} =0$

where

$\nabla \times$ is the curl operator

$\nabla \cdot$ is the divergence operator

$\mu _{0}\$ is the free-space permeability

$\mathbf {J} \$ is current density

$\partial \$ is the partial derivative

$\epsilon _{0}\$ is the free-space permittivity

$\mathbf {E} \$ is the electric field

$t\$ is time

The first equation is known as Ampère's law with Maxwell's correction. The second term of this equation (Maxwell's correction) disappears in static (time-independent) systems. The second equation is a statement of the observed non-existence of magnetic monopoles. These are two of the four Maxwell's equations written in the differential notation introduced by Oliver Heaviside.

Energy in the magnetic field

The energy of a long (or toroidal) solenoid is given by:

$\mathrm {Energy} =L{\frac {I^{2}}{2}}$

If we divide the energy by the volume of the solenoid, the density of the magnetic field energy can be obtained:

$u={\frac {B^{2}}{2\mu }}$

For example, a magnetic field B of one tesla has an energy density about 398 kilojoules per cubic metre, and of 10 teslas, about 40 megajoules per cubic metre.

This is the same as the pressure produced by magnetic field, since pressure and energy density are essentially the same physical quantities and thus have the same units. Thus, a magnetic field of 1 tesla produces a pressure of 398 kPa (about 4 atmospheres), and 10 T about 40 Mpa (~400 atm).

Symbols and terminology

Magnetic field is usually denoted by the symbol $\mathbf {B} \$. Historically, $\mathbf {B} \$ was called the magnetic flux density or magnetic induction. A distinct quantity, $\mathbf {H}$, was called the magnetic field (strength), and this terminology is still often used to distinguish the two in the context of magnetic materials (non-trivial permeability μ). Otherwise, however, this distinction is often ignored, and both quantities are frequently referred to as "the magnetic field." (Some authors call H the auxiliary field, instead.) In linear materials, such as air or free space, the two quantities are linearly related:

$\mathbf {B} =\mu \mathbf {H} \$

where

$\ \mu$ is the magnetic permeability of the medium, measured in henries per metre.

In SI units, $\mathbf {B} \$ and $\mathbf {H} \$ are measured in teslas (T) and amperes per metre (A/m), respectively; or, in cgs units, in gauss (G) and oersteds (Oe), respectively. Two parallel wires carrying an electric current in the same direction will generate a magnetic field that will cause a force of attraction between them. This fact is used to define the value of an ampere of electric current. While like charges repel and unlike ones attract, the opposite holds for currents: if the current in one of the two parallel wires is reversed, the two will repel.

Properties

Maxwell did much to unify static electricity and magnetism, producing a set of four equations relating the two fields. However, under Maxwell's formulation, there were still two distinct fields describing different phenomena. It was Albert Einstein who showed, using special relativity, that electric and magnetic fields are two aspects of the same thing (a rank-2 tensor), and that one observer may perceive a magnetic force where a moving observer perceives only an electrostatic force. Thus, using special relativity, magnetic forces are a manifestation of electrostatic forces of charges in motion and may be predicted from knowledge of the electrostatic forces and the velocity of movement (relative to some observer) of the charges.

A changing magnetic field is mathematically the same as a moving magnetic field (see relativity of motion). Thus, according to Einstein's field transformation equations (that is, the Lorentz transformation of the field from a proper reference frame to a non-moving reference frame), part of it is manifested as an electric field component. This is known as Faraday's law of induction and is the principle behind electric generators and electric motors.

Magnetic field lines

Magnetic field lines shown by iron filings

The direction of the magnetic field vector follows from the definition above. It coincides with the direction of orientation of a magnetic dipole, such as a small magnet, a small loop of current in the magnetic field, or a cluster of small particles of ferromagnetic material (see figure).

Pole labelling confusions

The end of a compass needle that points north was historically called the "north" magnetic pole of the needle. Since dipoles are vectors and align "head to tail" with each other, the magnetic pole located near the geographic North Pole is actually the "south" pole.

The "north" and "south" poles of a magnet or a magnetic dipole are labelled similarly to north and south poles of a compass needle. Near the north pole of a bar or a cylinder magnet, the magnetic field vector is directed out of the magnet; near the south pole, into the magnet. This magnetic field continues inside the magnet (so there are no actual "poles" anywhere inside or outside of a magnet where the field stops or starts). Breaking a magnet in half does not separate the poles but produces two magnets with two poles each.

Earth's magnetic field is produced by electric currents in its liquid core.

Field density

Magnetic field density, otherwise known as magnetic flux density, is essentially what the layman knows as a magnetic field —akin to a gravitational or electric field. It is a response of a medium to the presence of a magnetic field. The SI unit of magnetic flux density is the tesla. 1 tesla = 1 weber per square metre.

It can be more easily explained if one works backwards from the equation:

$B={\frac {F}{IL}}\,$

where

B is the magnitude of flux density, measured in teslas

F is the force experienced by a wire, measured in newtons

I is the current, measured in amperes

L is the length of the wire, measured in metres

File:Lhr.gif

Demonstration of Fleming's left hand rule

For a magnetic flux density to equal 1 tesla, a force of 1 newton must act on a wire of length 1 metre carrying 1 ampere of current.

1 newton of force is not easily accomplished. For example: the most powerful superconducting electromagnets in the world have flux densities of 'only' 20 T. This is true obviously for both electromagnets and natural magnets, but a magnetic field can only act on moving charge—hence the current, I, in the equation.

The equation can be adjusted to incorporate moving single charges, ie protons, electrons, and so on via

$F=BQv\,$

where

Q is the charge in coulombs, and

v is the velocity of that charge in metres per second.

Fleming's left hand rule for motion, current and polarity can be used to determine the direction of any one of those from the other two, as seen in the example. It can also remembered in the following way. The digits from the thumb to second finger indicate 'Force', 'B-field', and 'I(Current)' respectively, or F-B-I in short. For professional use, the right hand grip rule is used instead which originated from the definition of cross product in the right hand system of coordinates.

Other units of magnetic flux density are

1 gauss = 10^-4 teslas = 100 microteslas (µT)

1 gamma = 10^-9 teslas = 1 nanotesla (nT)

Rotating magnetic fields

Main article: Alternator

The rotating magnetic field is a key principle in the operation of alternating-current motors. A permanent magnet in such a field will rotate so as to maintain its alignment with the external field. This effect was utilised in early alternating-current electric motors. A rotating magnetic field can be constructed using two orthogonal coils with 90 degrees phase difference in their AC currents. However, in practice such a system would be supplied through a three-wire arrangement with unequal currents. This inequality would cause serious problems in standardization of the conductor size and so, in order to overcome it, three-phase systems are used where the three currents are equal in magnitude and have 120 degrees phase difference. Three similar coils having mutual geometrical angles of 120 degrees will create the rotating magnetic field in this case. The ability of the three-phase system to create a rotating field, utilized in electric motors, is one of the main reasons why three-phase systems dominate the world's electrical power supply systems.

Because magnets degrade with time, synchronous motors and induction motors use short-circuited rotors (instead of a magnet) following the rotating magnetic field of a multicoiled stator. The short-circuited turns of the rotor develop eddy currents in the rotating field of the stator, and these currents in turn move the rotor by the Lorentz force.

In 1882, Nikola Tesla identified the concept of the rotating magnetic field. In 1885, Galileo Ferraris independently researched the concept. In 1888, Tesla gained U.S. patent 381,968 for his work. Also in 1888, Ferraris published his research in a paper to the Royal Academy of Sciences in Turin.

Hall effect

Main article: Hall effect

Because the Lorentz force is charge-sign-dependent (see above), it results in charge separation when a conductor with current is placed in a transverse magnetic field, with a buildup of opposite charges on two opposite sides of conductor in the direction normal to the magnetic field, and the potential difference between these sides can be measured.

The Hall effect is often used to measure the magnitude of a magnetic field as well as to find the sign of the dominant charge carriers in semiconductors (negative electrons or positive holes).

Magnetic field of celestial bodies

A rotating body of conductive gas or liquid develops self-amplifying electric currents, and thus a self-generated magnetic field, due to a combination of differential rotation (different angular velocity of different parts of body), Coriolis forces and induction. The distribution of currents can be quite complicated, with numerous open and closed loops, and thus the magnetic field of these currents in their immediate vicinity is also quite multitwisted. At large distances, however, the magnetic fields of currents flowing in opposite directions cancel out and only a net dipole field survives, slowly diminishing with distance. Because the major currents flow in the direction of conductive mass motion (equatorial currents), the major component of the generated magnetic field is the dipole field of the equatorial current loop, thus producing magnetic poles near the geographic poles of a rotating body.

The magnetic fields of all celestial bodies are more or less aligned with the direction of rotation. Another feature of this dynamo model is that the currents are AC rather than DC. Their direction, and thus the direction of the magnetic field they generate, alternates more or less periodically, changing amplitude and reversing direction, although still more or less aligned with the axis of rotation.

The Sun's major component of magnetic field reverses direction every 11 years (so the period is about 22 years), resulting in a diminished magnitude of magnetic field near reversal time. During this dormancy time, the sunspots activity is maximized (because of the lack of magnetic braking on plasma) and, as a result, massive ejection of high energy plasma into the solar corona and interplanetary space takes place. Collisions of neighboring sunspots with oppositely directed magnetic fields result in the generation of strong electric fields near rapidly disappearing magnetic field regions. This electric field accelerates electrons and protons to high energies (kiloelectronvolts) which results in jets of extremely hot plasma leaving Sun's surface and heating coronal plasma to high temperatures (millions of K).

Compact and fast-rotating astronomical objects (white dwarfs, neutron stars and black holes) have extremely strong magnetic fields. The magnetic field of a newly born fast-spinning neutron star is so strong (up to 10⁸ teslas) that it electromagnetically radiates enough energy to quickly (in a matter of few million years) damp down the star rotation by 100 to 1000 times. Matter falling on a neutron star also has to follow the magnetic field lines, resulting in two hot spots on the surface where it can reach and collide with the star's surface. These spots are literally a few feet (about a metre) across but tremendously bright. Their periodic eclipsing during star rotation is believed to be the source of pulsating radiation (see pulsars).

Jets of relativistic plasma are often observed along the direction of the magnetic poles of active black holes in the centers of young galaxies.

If the gas or liquid is very viscous (resulting in turbulent differential motion), the reversal of the magnetic field may not be very periodic. This is the case with the Earth's magnetic field, which is generated by turbulent currents in a viscous outer core.

See also

General

• Electric field - effect produced by an electric charge that exerts a force on charged objects in its vicinity.
• Electromagnetic field - a field composed of two related vector fields, the electric field and the magnetic field.
• Electromagnetism - the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field.
• Magnetism - phenomenon by which materials exert an attractive or repulsive force on other materials.
• Magnetohydrodynamics - the academic discipline which studies the dynamics of electrically conducting fluids.
• Magnetic flux
• SI electromagnetism units

Mathematics

• Ampère's law - magnetic equivalent of Gauss's law.
• Biot-Savart law - describes the magnetic field set up by a steadily flowing line current.
• Magnetic helicity - extent to which a magnetic field "wraps around itself".
• Maxwell's equations - four equations describing the behavior of the electric and magnetic fields, and their interaction with matter.

Applications

• Helmholtz coil - a device for producing a region of nearly uniform magnetic field.
• Maxwell coil - a device for producing a large volume of almost constant magnetic field.
• Earth's magnetic field - a discussion of the magnetic field of the Earth.
• Dynamo theory - a proposed mechanism for the creation of the Earth's magnetic field.
• Electric motor - AC motors used magnetic fields

References

Books

• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 013805326X.
• Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 047130932X.
• Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0716708108.

External articles

Information

• Nave, R., "Magnetic Field". HyperPhysics.
• "Magnetism", The Magnetic Field. theory.uwinnipeg.ca.
• Hoadley, Rick, "What do magnetic fields look like?" 17 July 2005.

Field density

• Jiles, David (1994). Introduction to Electronic Properties of Materials (1st ed.). Springer. ISBN 0-412-49580-5.

Rotating magnetic fields

• "Rotating magnetic fields". Integrated Publishing.
• "Introduction to Generators and Motors", rotating magnetic field. Integrated Publishing.
• "Induction Motor-Rotating Fields".

Diagrams

• McCulloch, Malcolm,"A2: Electrical Power and Machines", Rotating magnetic field. eng.ox.ac.uk.
• "AC Motor Theory" Figure 2 Rotating Magnetic Field. Integrated Publishing.

Journal Articles

• Yaakov Kraftmakher, "Two experiments with rotating magnetic field". 2001 Eur. J. Phys. 22 477-482.
• Bogdan Mielnik and David J. Fernández C., "An electron trapped in a rotating magnetic field". Journal of Mathematical Physics, February 1989, Volume 30, Issue 2, pp. 537-549.
• Sonia Melle, Miguel A. Rubio and Gerald G. Fuller "Structure and dynamics of magnetorheological fluids in rotating magnetic fields". Phys. Rev. E 61, 4111–4117 (2000).